Convergence of the Rayleigh—Ritz method in self-consistent-field and multiconfiguration self-consistent-field calculations

We investigate the analytical convergence of SCF and MCSCF calculations, when the dimension of the subspaces to which the orbitals are restricted tends to infinity. We show that the completeness only inL ²(R ³;C ²) of the orbital bases does not ensure the convergence of the Ritz-energy, neither in SCF nor in MCSCF calculations, but that this convergence — as well as the convergence of the Ritz-orbitals in SCF calculations — is on the contrary guaranteed if the orbital bases are complete in the Sobolev spaceW ^1,2(R ³;C ²). Some consequences on the choice of the orbital exponents of Slater and Gauss functions are also discussed.     

An efficient ab initio gradient program

An ab initio Hartree-Fock gradient program is described. It is characterized by (1) efficiency of the gradient evaluation, and (2) capability of handling higher angular momentum (d andf) basis functions. The latter are constructed from shifted Cartesian Gaussian p-type primitives. A satisfactory solution is presented for the problems connected with the neglect of small integrals in a gradient program. Methods for increasing the efficiency of the SCF procedure are discussed.     

Analytical first derivatives of the energy for small CI expansions

Using the Z-vector formalism the analytical gradient of the energy in small CI expansions is derived and implemented for semiempirical MNDO-type methods. The computation time is shown to scale as O(N ³) with the size of the system, with the memory requirements growing as O(N ²). The evaluation of the analytical gradient is significantly faster than the underlying SCF and CI calculations, so that routine full geometry optimizations at the semiempirical CI level become possible for large systems. Received: 24 April 1997 / Accepted: 10 June 1997     

A general ab initio molecular multi-configuration self-consistent field algorithm

The ab initio multiconfiguration self-consistent-field (MC SCF ) techniques and computer programs of Basch [1, 2] and the ab initio configuration interaction (CI ) techniques and programs of Whitten and Hackmeyer [3] have been combined and generalized to form a general technique and program to yield optimized ab initio MC SCF wavefunctions for any set of Slater determinants. The Slater determinants are read in as input data to the program along with the spin parity that is being considered (optional) and the program successively does the CI calculation and one iteration of the SCF calculation, constructing the proper Fock–Hamiltonians by examining the set of Slater determinants and their CI coefficients. The Fock–Hamiltonian matrices are calculated and diagonalized in succession, a single two-dimensional array being used to store these matrices. The basis function integrals are read from a tape only once during each MC SCF iteration (one MC SCF iteration = a CI calculation followed by one iteration of the SCF calculation).     

Ab-initio and approximately rigorous calculations on small,medium, and large systems

We have explored two areas of approximately rigorous calculations for computing nonempirical wave functions for heavy and/or large molecules orders of magnitude faster than with conventional ab-initio methods but with the same chemical accuracy. First, we have developed and used a series of programs (starting from our new fast sets of ab-initio Gaussian SCF and SCF -CI programs) incorporating ab-initio effective core model potentials (MOD -POT ) which allow one to treat only the valence electrons explicitly, plus a charge conserving integral prescreening, which cuts down significantly on the number of integrals that have to be calculated, stored, or processed for a large molecule. We have named this latter procedure VRDDO (variable retention of diatomic differential overlap). With these MODPOT and MODPOT /VRDDO methods we have explored a variety of small, medium, and large systems ranging from electron affinities of atoms through to molecules of biological interest and large boron hydrides. The results compared to ab-initio SCF or SCF /CI calcuations are very good, usually within 0.001 to 0.002 a.u. for orbital energies and gross atomic populations (GAPS ) and even better along potential energy curves. Secondly, we have explored the use of the MS -Xα method for less conventional molecules and properties than those for which it is customarily employed.     

Theoretical studies of molecular conformation. Derivation of an additive procedure for the computation of intramolecular interaction energies. Comparison withab initio SCF computations

An additive procedure (SIBFA) is developed for the rapid computation of conformational energy variations in very large molecules. The macromolecule is built out of constitutive molecular fragments and the intramolecular energy is computed as a sum of interaction energies between the fragments. The electrostatic and the polarization components are calculated using multicenter multipole expansions of theab initio SCF electron density of the fragments. The repulsion component is obtained as a sum of bond and lone pair interactions.Tests of the procedure on a series of model compounds containing ether oxygens and pyridine-like nitrogens are reported and compared with the results of correspondingab initio SCF calculations. The resulting methodology is compatible with the simultaneous computation of intermolecular interactions.     

On the SCF calculation of excited states: Singlet states in the two-electron problem

The problem of determining SCF wave functions for excited electronic states is examined for singlet states of two-electron systems using a Lowdin natural orbital transformation of the full CI wave function. This analysis facilitates the comparison of various SCF methods with one another. The distribution of the full CI states among the natural orbital MCSCF states is obtained for the S states of helium using a modest Gaussian basis set. For SCF methods that are not equivalent to the full CI wave functions, it is shown that the Hartree-Fock plus all single excitation wave functions are equivalent to that of Hartree-Fock plus one single excitation. It is further shown that these wave functions are equivalent to the perfect pair or TCSCF wave functions in which the CI expansion coefficients are restricted to have opposite signs. The case of the natural orbital MCSCF wave function for two orbitals is examined in greater detail. It is shown that the first excited state must always be found on the lower natural orbital MCSCF CI root, thus precluding the use of the Hylleras-Undeim-MacDonald (HUM) theorem in locating this state. It is finally demonstrated that the solution obtained by applying the HUM theorem (minimizing the upper MCSCF CI root with respect to orbital mixing parameters) is an artifact of the MCSCF method and does not correspond to any of the full CI states.     

Optimization of equilibrium geometries and transition structures

A modified conjugate gradient algorithm for geometry optimization is outlined for use with ab initio MO methods. Since the computation time for analytical energy gradients is approximately the same as for the energy, the optimization algorithm evaluates and utilizes the gradients each time the energy is computed. The second derivative matrix, rather than its inverse, is updated employing the gradients. At each step, a one-dimensional minimization using a quartic polynomial is carried out, followed by an n-dimensional search using the second derivative matrix. By suitably controlling the number of negative eigenvalues of the second derivative matrix, the algorithm can also be used to locate transition structures. Representative timing data for optimizations of equilibrium geometries and transition structures are reported for ab initio SCF –MO calculations.     

Static and dynamic density functional investigation of hydrated beryllium dications

Using density functional theory with Becke's gradient correction for the exchange part and Vanderbilt's ultrasoft pseudopotentials, we investigated [Be(H₂O)_n]²⁺ clusters for n = 1 and 3. A new scheme implemented with the Car-Parrinello method is used, upon which no periodic boundary conditions are imposed so that isolated and possibly charged molecules can also be treated dynamically in a plane-wave basis. Harmonic vibrational frequencies are obtained via a fit of the molecular dynamics trajectory in terms of harmonic oscillators, for which we use a fragmentation scheme to analyze complex spectra. Based on this combination of techniques, we find good agreement with data from SCF calculations and correlated methods. © 1996 John Wiley & Sons, Inc.     

Molecular electric properties in electronic excited states: multipole moments and polarizabilities of H2O in the lowest1 B 1 and3 B 1 excited states

Summary The dipole and quadrupole moments and the dipole polarizability tensor components are calculated for the¹ B ₁ and³ B ₁ excited states of the water molecule by using the complete active space (CAS) SCF method and an extended basis set of atomic natural orbitals. The dipole moment in the lowest¹ B ₁ (0.640 a.u.) and³ B ₁ (0.416 a.u.) states is found to be antiparallel to that in the ground electronic state of H₂O. The shape of the quadrupole moment ellipsoid is significantly modified by the electronic excitation to both states investigated in this paper. All components of the excited state dipole polarizability tensor increase by about an order of magnitude compared to their values in the ground electronic state. The present results are used to discuss some aspects of intermolecular interactions involving molecules in their excited electronic states.     

Parallel direct SCF and gradient program for workstation clusters

A parallel direct SCF and gradient program for workstation clusters has been implemented on the basis of the ab initio program package TURBOMOLE. Applications on large molecular systems monitor an appreciable speedup in residence time. © John Wiley & Sons, Inc.     

Ab initio calculation of the dipole moment function of hydrogen iodide

SCEP/CEPA and MC SCF potential energy and dipole moment functions for hydrogen iodide have been calculated. Spectroscopic constants and vibrational dipole matrix elements obtained from the CEPA functions are in good agreement with experimental data. In contrast to previous results for hydrogen fluoride, the MC SCF dipole moment function is less accurate than the CEPA function.     

环己硅烷类液晶化合物的量子化学研究: 联苯基乙烷类系列

运用AM1和PM3两种SCF-MO方法, 通过能量梯度全优化计算, 给出了25种环己硅烷类液晶化合物的稳定几何构型、电子结构和生成热、偶极矩等基本性质。联系有机电子结构理论进行了讨论。     

Modified one-electron hamiltonian method in the MC SCF theory

For orbital optimization within the MC SCF theory a modification of the OEH method is proposed with the direction of descent determined according to the Fletcher–Reeves gradient method. The combined method developed on this basis ensures the convergence of the iterative process when the Hessian singularities occur. The convergence properties of the method proposed are studied by performing the ab initio water molecule calculations using two types of multiconfigurational wave functions.     

Transition structures and energy barriers of pericyclic reactions in the CASSCF approach

Five energy hypersurfaces of the most examined pericyclic reactions have been investigated by using ab initio SCF , CASSCF , and the semiempirical AM 1 methods. The systems are H₄, H₆, C₃H₆, C₄H₄, and C₃OH₄. Stationary points and their sets of harmonic vibrational frequencies have been calculated by means of analytical gradient techniques in the frameworks of the respective approaches. ZPE corrected energy barriers are based on single-point calculations including dynamical correlation corrections by CAS -CI (SD )+DC , CASCEPA , or MP 2.     

Gradient and mc scf calculations of the conformers and the formation of primary ethylene ozonide

The confonners of primary ethylene ozonide have been studied by ab initio gradient and MC SCF calculations. At the MC SCF level they are more spread in energy than in SCF calculations. The planar conformer, carbon-carbon half chair and the oxygen envelope are much higher m energy than the other conformers. The MC SCF activation energy for cyclo-addition of ozone and ethylene is 91–99 $\text{kJ}\text{mole}$.     

Internal orbitally resolved charge sensitivity analysis

The intramolecular (internal) charge sensitivity analysis (ICSA ) is presented in the molecular orbital (MO ) resolution, with diagonal idempotency constraints being imposed on the MO occupations, n, which can then be considered as independent electron population variables of the system for the assumed fixed shapes of MOS . The standard closed-shell (C.S.) SCF MO framework is adopted with the energy function E(n) approximated by the average energy of a configuration expression to cover fractional MO occupations. The components of the gradient of the auxiliary energy function, including the constraint terms, with respect to n, identified as negative MO internal electron potentials, vanish identically for the ground-state c.s. configuration, whereas their responses to hypothetical intramolecular charge displacements assure the system stability. An application of the ICSA to the iterative SCF procedure is suggested. Illustrative results for the water molecule are reported and general properties of molecular normal orbitals (eigenvectors of the hardness matrix) are discussed. © 1992 John Wiley & Sons, Inc.     

Automation of pH Optimization Experiments During LC Development

The selection of an optimal mobile phase pH under solvent gradient conditions is experimentally challenging. Although quaternary pumps are widely available, they are often used in binary mode to run simple solvent gradients with one pH-adjusted buffer at a time. A more effective use of quaternary pumps is to deliver two different aqueous buffer components (A and B) in a constant proportion to simulate a single, premixed buffer component, while simultaneously producing a solvent gradient by increasing the organic solvent component (S). This approach largely automates the pH optimization experiments. A more detailed investigation of pH effects becomes possible with less time and effort. Once a suitable pH has been identified, the same separation can be reproduced by a simpler binary gradient method which is more suitable for routine work. This study demonstrates the feasibility of this approach both theoretically and through actual examples.     

1-异丙基环己硅烷类液晶化合物的量子化学研究--取代联苯基乙烷系列

我们曾用半经验的分子轨道方法对标题化合物的某些系列进行过计算[1 ,2 ] 。本文选择 7种 1 异丙基环己硅烷 取代联苯基乙烷系列化合物 ,以ＳＣＦ—ＭＯ—ＡＭ1 [3] 和ＰＭ3[4] 两种方法 ,优化出它们稳定的几何构型 (分子的总能量最低、核—核排斥能最小 )并对其作了振动分析 ;同时计算了标题物在该稳定构型下的电子结构和若干基本性质。1　计算图 1示出了 7种标题化合物的结构及原子编号。用ＭＯＰＡＣ7.0程序中的ＡＭ1 和ＰＭ3方法 ,分子的初始构型参数采用Ｐｏｐｌｅ’ｓ标准构型数据[5] 。对每个分子实行能量梯度全优化和ＳＣＦ计算。…